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Distributions are of two types: those that are obtained from locally integrable functions, and those that aren't. For the first type, the support of distribution is simply the support of the function. For the other kind of distribution, for example, the Dirac Delta 'function', we can't find the support this way.

The support of the Dirac Delta distribution is given to be the set $\{0\}$. Can someone help me in understanding why?

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    $\begingroup$ Read this: en.wikipedia.org/wiki/… It explains how the support of a distribution is defined in the general case, with the example of $\delta$. $\endgroup$ – Augustin Aug 6 '15 at 15:25
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    $\begingroup$ It basically has zero support because it is used so improperly. $\endgroup$ – Disintegrating By Parts Aug 6 '15 at 17:02
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Well the definition of the support is defined as the part of the domain where the distribution is non-zero. There are a few definitions of the Dirac delta but all agree that $x\neq 0 \implies \delta(x) = 0$.

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    $\begingroup$ You're implying that for all definitions of $\delta(x)$, the expression $\delta(x)$ makes sense for a fixed $x \in \Bbb R$ (for example, that we could say something like $\delta(1) = 0$). This is not the case. $\endgroup$ – Ben Grossmann Aug 6 '15 at 15:30
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    $\begingroup$ That's not true. Rigorous treatments of the Dirac delta do not refer to any pointwise behavior. $\endgroup$ – Cameron Williams Aug 6 '15 at 15:30
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    $\begingroup$ The Dirac Delta is NOT a function. Rather it is a Generalized Function, or Distribution. Therefore, it is not a mapping that is assigned a value. $\endgroup$ – Mark Viola Aug 6 '15 at 15:39

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